3.226 \(\int (e x)^m \cot (d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=100 \[ \frac {i (e x)^{m+1}}{e (m+1)}-\frac {2 i (e x)^{m+1} \, _2F_1\left (1,-\frac {i (m+1)}{2 b d n};1-\frac {i (m+1)}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (m+1)} \]

[Out]

I*(e*x)^(1+m)/e/(1+m)-2*I*(e*x)^(1+m)*hypergeom([1, -1/2*I*(1+m)/b/d/n],[1-1/2*I*(1+m)/b/d/n],exp(2*I*a*d)*(c*
x^n)^(2*I*b*d))/e/(1+m)

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Rubi [F]  time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (e x)^m \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(e*x)^m*Cot[d*(a + b*Log[c*x^n])],x]

[Out]

Defer[Int][(e*x)^m*Cot[d*(a + b*Log[c*x^n])], x]

Rubi steps

\begin {align*} \int (e x)^m \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int (e x)^m \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}

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Mathematica [A]  time = 13.67, size = 182, normalized size = 1.82 \[ -\frac {i x (e x)^m \left (\frac {(m+1) e^{2 i a d} \left (c x^n\right )^{2 i b d} \, _2F_1\left (1,-\frac {i (m+2 i b d n+1)}{2 b d n};-\frac {i (m+4 i b d n+1)}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{2 i b d n+m+1}+\, _2F_1\left (1,-\frac {i (m+1)}{2 b d n};1-\frac {i (m+1)}{2 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*Cot[d*(a + b*Log[c*x^n])],x]

[Out]

((-I)*x*(e*x)^m*(Hypergeometric2F1[1, ((-1/2*I)*(1 + m))/(b*d*n), 1 - ((I/2)*(1 + m))/(b*d*n), E^((2*I)*d*(a +
b*Log[c*x^n]))] + (E^((2*I)*a*d)*(1 + m)*(c*x^n)^((2*I)*b*d)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m + (2*I)*b*
d*n))/(b*d*n), ((-1/2*I)*(1 + m + (4*I)*b*d*n))/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/(1 + m + (2*I)*b*
d*n)))/(1 + m)

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \cot \left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*cot(d*(a+b*log(c*x^n))),x, algorithm="fricas")

[Out]

integral((e*x)^m*cot(b*d*log(c*x^n) + a*d), x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*cot(d*(a+b*log(c*x^n))),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 1.88, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \cot \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*cot(d*(a+b*ln(c*x^n))),x)

[Out]

int((e*x)^m*cot(d*(a+b*ln(c*x^n))),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*cot(d*(a+b*log(c*x^n))),x, algorithm="maxima")

[Out]

integrate((e*x)^m*cot((b*log(c*x^n) + a)*d), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,{\left (e\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*(a + b*log(c*x^n)))*(e*x)^m,x)

[Out]

int(cot(d*(a + b*log(c*x^n)))*(e*x)^m, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \cot {\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*cot(d*(a+b*ln(c*x**n))),x)

[Out]

Integral((e*x)**m*cot(a*d + b*d*log(c*x**n)), x)

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